Question

A biotechnology laboratory operates two automated synthesis workflows, System X and System Y, to manufacture two compounds, Alpha ($A$) and Beta ($B$). System X operates for $x$ hours, producing $A$ at a rate of $3\text{ mg/hr}$ and $B$ at a rate of $5\text{ mg/hr}$. System Y operates for $y$ hours, producing $A$ at a rate of $15\text{ mg/hr}$ and $B$ at a rate of $8\text{ mg/hr}$. The combined mass of compounds $A$ and $B$ produced across both workflows is exactly $340\text{ mg}$. Furthermore, the system's operating efficiency requires that the ratio of operating hours $\frac{x}{y}$ equals the ratio of total mass produced $\frac{B}{A}$. What is the total operating time, $x + y$, in hours?